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Exploring Periodic Functions Through Ferris Wheel and Tide Examinations

Dive into the world of periodic functions with a focus on a Ferris wheel model and tide interpretations, including graph analysis, transformations, and equations. Explore cosine, sine functions, and more in this educational context.

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Exploring Periodic Functions Through Ferris Wheel and Tide Examinations

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  1. Math 170 Functions, Data, and Models 26 Periodic Functions Section 7.1-3 Exam 2 Take Home due Need graph paper and compasses.

  2. Review of Familiar Functions • Graph . (Additive processes) • Graph . (Multiplicative processes) • Graph . (Scaling) • Graph .

  3. Ferris Wheel • A Ferris wheel of diameter 50 meters completes one revolution every 2 minutes. You start at the lowest point on the wheel, which is still 5 meters off the ground. • Draw a scale model of the ground and Ferris wheel. • Construct a table of your height at 15 second intervals during one revolution of the wheel. • Draw a graph of height as a function of time from the lowest point. Include 3 revolutions of the wheel. • Determine the midline, amplitude, and period of the function.

  4. Simplified Tide • Interpret the point identified by the red arrow. • Is the tide rising or falling at 7:00 AM? • When does low tide occur? • What is the midline? • What is the amplitude? • What is the period?

  5. Transformation • Let be the inches above mean sea level at hours since 6:00 AM. • Let be the meters above ground at minutes since the lowest point. • Describe in terms of .

  6. Cosine and Sine • Angles are measured counterclockwise from the positive horizontal with 360 degrees corresponding to one full revolution. • If is the point of the unit circle specified by angle , then and . • Draw a unit circle: center at (0, 0) and radius 1. • Construct a table of cosine and sine at 0, 30, 60, 90, …, 330 degrees. • Draw graphs of cosine and sine on the domain 0 to 720 degrees. • Determine the midline, amplitude, and period of the function. • Estimate and calculate , , , and . • Find , , , and exactly.

  7. Transforming Cosine and Sine • In the following, assume that the inputs to cosine and sine are in degrees. • Sketch a graph of . • Sketch a graph of . • Sketch a graph of .

  8. Equations • Find equations for these functions.

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